The Semigroup Complexes and the Endomorphism Semigroups
by
Vitaly Usenko
Mathematical Department Slovyans'k State Pedagogical Institute, Slovyans'k, Ukraine

Call [\rho:S, T:\rho] a double-semigroup (or semigroup pair as in [1, 2]) whenever (S, *), (T, *) are monoids, \rho:S --> T(T):x --> \rho_x, \lambda:T --> T(S):x --> \lambda^x are homomorphism and accordinly antihomomorphism such that conditions (t*u)\rho_x=t\rho_x\lambda^u*u\rho_x, (x*y)\lambda^t=x\lambda^t*y\lambda^t\rho_x, x, y in S, t, u in T are holds (T(X) is symmetric semigroup over set X). The categories of semigroup complexes in terms of double-semigroups are defined.

Free and universal object of these categories are described. Some new semigroup constructions are defined such as double-wreath holomorph HWr^X_Y M of monoid M over the sets X and Y, biholomorph Hol (M₁, M₂) of monoids M₁ and M₂. Some series of submonoids and congruences of this constructions are described (so named quasi-affine, affine submonoids and congruences etc.) This notions and constructions are applied to description of structure properties some semigroups of endomorphisms.

Let G[X] to be a free product of group G and of infinite cyclic < x > , Mg G[x] - the semigroup of so named Menger's endomorphisms j_w of G[x] ((u(x))j_w=u(w(x)), u(x), w(x)=w in G[x]).

Theorem. The semigroup Mg G[x] is embeddable in quasi-affine submonoid of biholomorph Hol (P_w[G], < x > ), where P_w[G] - free countable power of G.

Theorem ([3]). Let M⁰(I, G, J;P) - reqular Rees matrix semigroup over group G⁰ with sendwich-matrice P. Some affine submonoid M and affine congruence \kappa of double-wreath holomorph HWr^I_J G are exists such that

End  M0(I, G, J:P) =~ M/\kappa.

Let F(X) is free group over X  (|X| <= \aleph₀) , E(X)=End F(X), E₀(X)={ j in E(X)  |  Im j = < w > ,   w in F(X)}.

Theorem. E₀(X) is weakly reductive ideal of E(X).  E(X) is unique (up to isomorphism) maximal dense ideal extension of E₀(X).

This theorem and the description of E₀(X) [4] are developed of results from [5].

References.

1. Usenko V.M. On category of B.Neumann's semigroup pairs // Second Intern. algebr. conf. in Ukraine dedicated to the mem. prof. L.A.Kaloujnine. (Kyiv-Vinnitsia, 9-16 May 1999).- p.118-120 (in Russian).

2. Usenko V.M. Semigroups and near-rings of transformations // Manuscript. Thesis for doctor's degree. Kyiv Taras Schevchenko Univ., Kyiv.-1999 (in Ukrainian).

3. Usenko V.M. The endomorphisms of completely 0-simple semigroups // Problems in algebra. vol. 13.- Gomel: University-Press.- 1998.- p.93-120 (in Russian).

4. Usenko V.M. The endomorphisms of free groups / /Problems in algebra. Vol.14.-Gomel:Univ. Press.- 1999.- pp.166-172. (in Russian).

5. Gluskin L.M. Semigroups and rings of endomorphisms of linear spaces // Izv.AN SSSR.- 1959.- vol.23.- p.841-870 (in Russian)

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-37.